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Albanese Torsors and the Elementary Obstruction Than a Mere Comparison Between the Orders of Two Classes

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Albanese Torsors and the Elementary Obstruction Than a Mere Comparison Between the Orders of Two Classes ON ALBANESE TORSORS AND THE ELEMENTARY OBSTRUCTION OLIVIER WITTENBERG 1. Introduction If X is a principal homogeneous space, or torsor, under a semi-abelian variety A over a field k, one can naturally associate with it two integers which give a measure of its nontriviality: the index I(X), defined as the greatest common divisor of the degrees of the closed points of X, and the period P (X), which is the order of the class of X in the Galois cohomology group H1(k, A). The period divides the index. In the classical situation where A is an abelian variety, the extent to which the period and the index can fail to be equal has been investigated by a number of authors, including Lang and Tate (see [16]), Ogg, Cassels, Shafarevich, Lichtenbaum. On the other hand, the well-known period-index problem for central simple algebras, first raised by Brauer, is a particular case of the study of periods and indices of torsors under tori. Let X now denote an arbitrary smooth variety over k (not necessarily complete). One may define the index I(X) as above, and the period P (X) as the supremum of P (Y ) when Y ranges over all torsors under semi-abelian varieties such that there exists a morphism X → Y . As was proven by Serre, there is a universal object among morphisms from X to torsors under semi-abelian varieties: the Albanese 1 1 torsor X → AlbX/k. Hence P (X)= P (AlbX/k). Periods of open varieties turn out to be relevant even if one is primarily interested in complete varieties. Indeed the period of a small enough dense open subset U of a complete variety X often provides valuable information which is not given by the period of X alone. For instance, if X is the Severi-Brauer variety attached to a central simple algebra α, then 1 obviously P (X) = 1 (since AlbX/k is just a point), whereas P (U) can be shown to equal the exponent of α if U ⊆ X is small enough. Quite generally, if U ⊆ X is open dense, then P (U) divides I(X). Thus, if we define the generic period Pgen(X) as the supremum of P (U) when U ranges over all dense open subsets of X, then for a 0-cycle of degree 1 to exist on X, it is necessary that Pgen(X) = 1. Colliot-Thélène and Sansuc defined and studied in [5] another general obstruction to the existence of 0-cycles of degree 1, the so-called elementary obstruction. Namely, denoting by k a separable closure of k and by k(X) the function field arXiv:math/0611284v2 [math.AG] 16 Aug 2007 ⋆ ⋆ of X ⊗k k, the elementary obstruction is said to vanish if and only if the inclusion k ⊂ k(X) admits a Galois-equivariant retraction. The theme of the present paper is that these two obstructions are very tightly related. We shall prove in particular that Pgen(X) = 1 implies the vanishing of the elementary obstruction, over an arbitrary field k (Theorem 2.2). When k is a p-adic field, a real closed field, a number field, or a field of dimension ≤ 1 in the sense of Serre, we are able to obtain stronger statements. Most notably, the condition Pgen(X) = 1 is even equivalent to the vanishing of the elementary obstruction if k is a p-adic or real closed field (Theorem 3.2.1), or if k is a number field and the Tate-Shafarevich group of the Picard variety of X is finite (Theorem 3.3.1); we give several applications of these results in §3. On the Date: September 26, 2006; revised on August 15, 2007. 1 2 OLIVIER WITTENBERG other hand, if k is a field of dimension ≤ 1, we prove that the elementary obstruction always vanishes (Theorem 3.4.1), while it is well known that P (X) (and therefore Pgen(X)) may be greater than 1. The elementary obstruction to the existence of 0-cycles of degree 1 on X is known to vanish if and ⋆ only if the Yoneda equivalence class of a certain 2-extension e(X) of Pic(X ⊗k k) by k in the category of discrete Galois modules is trivial (see [5, Proposition 2.2.4]). One is naturally led to consider an analogous 2-extension E(X) of the relative Picard functor PicX/k by the multiplicative group Gm; it is a straightforward generalisation of the elementary obstruction. We devote the last section of this paper to showing that the 2-extension E(X) contains enough information to reconstruct, in a strong sense, the 1 Albanese torsor AlbU/k for any open U ⊆ X (Theorem 4.2.3). More precisely, for any open U ⊆ X, the M 2-extension E(X) gives rise, by pullback, to a 2-extension E (U) of the Picard 1-motive of U by Gm. 1 Theorem 4.2.3 essentially states that AlbU/k is the variety which parametrises, in an appropriate sense, “Yoneda trivialisations” of the 2-extension EM(U). In particular, the Yoneda equivalence class of the M 1 ∅ 2-extension E (U) is trivial if and only if AlbU/k(k) 6= , that is, if and only if P (U) = 1. From this we deduce, without any assumption on the field k, that the Yoneda equivalence class of E(X) is trivial if and only if Pgen(X) = 1 (Corollary 4.2.4). This is of course stronger than our previous assertion that the elementary obstruction vanishes if Pgen(X) = 1. M 1 Producing trivialisations of the 2-extension E (U) from rational points of AlbU/k (or from rational points of U) is not particularly difficult (see Proposition 4.1.6 and the proof of Corollary 4.2.4); it is M 1 however much less clear how to convert trivialisations of E (U) into rational points of AlbU/k. One of the key tools in the proof of Theorem 4.2.3 is a result which might be of some independent interest: we exhibit an explicit Poincaré sheaf on any abelian variety (Theorem 4.3.2). Acknowledgements. This work grew out of an attempt to answer the questions raised by Borovoi, Colliot-Thélène, and Skorobogatov in [1], and to get a better understanding of a result of Skoroboga- tov [35, Proposition 2.1]. In response to my queries about automorphisms of n-extensions, Shoham Shamir very kindly directed me to the papers of Retakh [27] and of Neeman and Retakh [23], and explained to me a variant of Lemma 4.4.1 below. I am grateful to Dennis Eriksson for our many stim- ulating discussions about Albanese torsors and generic periods; it should be noted that generic periods first appeared in [8]. Finally I am pleased to thank Jean-Louis Colliot-Thélène for his constant interest and encouragement, and Joost van Hamel for a number of useful comments and for pointing out the connections between §4 and [37]. Notation. Let k be a field. A variety over k is by definition a k-scheme of finite type. The Néron- Severi group NS(X) of a smooth k-variety X is the quotient of Pic(X) by the subgroup of all classes which are algebraically equivalent to 0 over k. If A0 is an algebraic group over k and A1 is a k-torsor under A0, we shall generally denote by An the n-fold contracted product of A1 with itself under A0 (so that the class of An in the Galois cohomology group H1(k, A0) is n times the class of A1). If A is a semi-abelian variety over k (that is, an extension of an abelian variety by a torus), we denote by TA the largest k-torus contained in A (so that A/TA is an abelian variety). Let A be an abelian category. We denote by C(A ) the category of complexes of objects of A . If f is a morphism in C(A ), we also denote by C(f) the mapping cone of f ([14, §1.4]); the context will make it clear which meaning is intended. n For n ∈ N and A, B ∈ A (resp. A, B ∈ C(A )), we denote by ExtA (A, B) the corresponding Ext (resp. hyperext) group. If A is the category of discrete modules over a ring R (resp. over a profinite n n n group Γ), we write ExtR(A, B) (resp. ExtΓ(A, B)) for ExtA (A, B). Finally, by a 1-motive we shall always mean a Deligne 1-motive, i.e., a 1-motive as defined in [26]. ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 3 2. The order of the elementary obstruction Let k be a field and X be a geometrically integral variety over k. The Albanese variety and the Albanese torsor of X over k, if they exist, are a semi-abelian variety 0 1 0 1 AlbX/k over k and a k-torsor AlbX/k under AlbX/k, endowed with a k-morphism uX : X → AlbX/k. They are characterised by the following universal property: for any semi-abelian variety A0 over k, any torsor A1 under A0, and any k-morphism m: X → A1, there exists a unique k-morphism of varieties 1 1 1 1 f : AlbX/k → A such that f ◦ uX = m, and there exists a unique k-morphism of algebraic groups 0 0 0 1 0 f : AlbX/k → A such that f is f -equivariant. Obviously, the Albanese variety, the Albanese torsor, and the morphism uX are unique up to a unique isomorphism.
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